# Bifurcation for Minimal Surface Equation in Hyperbolic $3$-Manifolds

## TL;DR

This paper investigates the existence, multiplicity, and asymptotic behavior of minimal surface immersions in hyperbolic 3-manifolds, revealing conditions for uniqueness, multiple solutions, and blow-up phenomena, with extensions to prescribed extrinsic curvature.

## Contribution

It provides new criteria for solution uniqueness, analyzes blow-up behavior based on genus, and extends the theory to prescribed extrinsic curvature problems.

## Key findings

- Unique solutions determined by conformal structure and second fundamental form
- Multiple solutions can occur depending on geometric conditions
- Blow-up behavior varies with the genus of the surface

## Abstract

Initiated by the work of Uhlenbeck in late 1970s, we study questions about the existence, multiplicity and asymptotic behavior for minimal immersions of closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions. In particular, we determine when exactly the solution is unique and when multiple solutions appear. Moreover, we analyze in detail the asymptotic behavior of the solutions when (and how) blowing up might occur. Interestingly the blow-up analysis exhibit different behaviors when the surface is of genus two or greater. Furthermore, we extend this program to consider similar problems where the total extrinsic curvature is prescribed and we prove an existence result.